6
In this chapter there is an explanation about how the optimization problem was posed and how the costs were estimated. It also presents the compression train superstructure implementation pro- cedure and associated simplifications and assumptions.
6.1 MINLP Formulation
Process Optimization is usually a mathematical systematic procedure based on the models cre- ated to describe systems. This action avoids the manual changing of the decision variable values by running several times the same simulation using the so called trial and error methodology. One of the main drawbacks is that, even for small problems, it is difficult to manually satisfy all process constraints or guarantee of a real optimality. So, in order to make sure all constraints are satisfied and guarantee that the solution obtained is the global optimal solution, the problem needs formal, mathematically-based methods for optimizing steady-state and dynamic process behavior.
Once the process model is ready and operational, all the degrees of freedom (DOF) are known, so it is simple to formulate an optimization problem, decision variables and all the constraints needed to solve it. After formulating the problem, gPROMS can perform both dynamic and steady-state optimization, subject to wide range of constraints, and both continuous and discrete decision variables can be optimized.
Generally, the optimization framework can be defined as:
minu (φ) = Z tf
0
zdt (6.1)
Where φis the objective function (what we want to optimize), u is the vector of parameterized control signals (what conditions and DOF we want to change to get to the result) andtf is the time horizon and time intervals for parameterization of the control variables.
These function is subjected to:
f(˚x, x, y, u, p) = 0 (6.2)
g(˚x, x, y, u, p)≤0 (6.3)
Wheref represents the equality constraints andg represents the inequality constraints. xis the vector of the state variables (assigned DOF),˚xis the derivative ofx,yis the vector of the algebraic variables (time-invariant variables) andpis the parameters’ vector.
In this case, it is a steady-state problem and the objective function is the total cost (Ctot) of the compression train, i.e, the sum of the annualized Capital Expenditure (CAPEX) (by dividing total capital cost by the number of project yearsNyear) and Operating Expenditure (OPEX). The decision variables are the number of compression sections (Ns) and the pressure ratiopr(j)of each sectionj.
Ctot(Ns, pr(j)) = CAP EX(Ns, pr(j)) Nyear
+OP EX(Ns, pr(j)) (6.4) These function is subjected to two constraints: the discharge temperature of each compressor cannot be bigger than 180oC (453.15 K) and the discharge pressure of the last compressor section
has to be between 30 and 40 bar (operating range of the dehydrator).
Tdisch(j)≤453.15 (6.5)
30≤pdisch(Ns)≤40 (6.6)
Note that, sinceNsis an integer variable, the problem has both continuous and integer decision variables. Also, due to the nature of the equation system having a non-linear behavior, it makes the problem into a Mixed Integer Non-Linear Programming (MINLP).
Often the corresponding MINLP models exhibit special structures (e.g. graphs, networks, sep- arable functions) that can be effectively exploited for developing specialized solution procedures.
However, it is also very often the case, particularly in engineering design, that nonlinearities in the continuous variables do not exhibit any special form since they result from complex engineering mod- els.
In order to solve these problems, there are many different general-purpose algorithms, like the branch and bound (Beale, 1977; Gupta, 1980) and the Outer Approximation/Equality Relaxation (OA/ER) Method (Duran and Grossmann, 1986; Kocis and Grossmann, 1987) [54].
The branch and bound and OA/ER algorithms require that some form of convexity assumption be satisfied in order to guarantee than they can find the global optimum of the MINLP. On the other hand, the OA/ER algorithm, wich tends to be the most efficient method when the Non-Linear Programming (NLP) subproblems are expensive or difficult to solve, is the most stringent in terms of convexity requirements. However, gPROMS uses is the OA/ER solver but with some modifications described in [54] called OAERAP.
Grossmann proposes that this new variant of the OA/ER algorithm, which does not require the ex- plicit identification of nonconvexities, by incorporating an augmented penalty function for the violation of linearizations of the nonlinear functions in the MINLP master problem.
The OAERAP algorithm decomposes the MINLP into a NLP sub-problem and a Mixed Integer Linear Programming (MILP) master problem. First it solves the NLP relaxation of the integer variables (in this case, the number of compression sections can have a non-integer value) to obtain the first intermediate iteration to the next problem. After that, the MILP master problem finds an integer point that features an augmented penalty function to find the minimum over the convex linearized function.
Then, it solves a NLP, fixing the integer variables, to find the optimum value of the continuous variables (pressure ratio of each compression section). Finally, it calculates the gradient based on the linearized functions and determines if the optimal point was reached or rather it needs to do another iteration and calculates the respective point.
Note that, due to the linearization of the non-convex functions, there is no guarantee of finding the global optimum. Since the compression train system is very non-linear, this may cause difficulties to find the global minimum of the total cost. However, it is likely that, if a good initial guess is provided, the global optimum can be found since the closer the initial guess is to the optimal solution the easier it is to find it.
6.2 Objective Function and Cost Estimation
6.2.1 Objective Function
As it was referred in section 6.1, the objective function to be minimized is the total cost of the compression system based on the sum between the annualized Capital Expenditure (CAPEX) and Operating Expenditure (OPEX), as defined in equation 6.4. In CAPEX, the following costs were considered: compressor sections, coolers, electric drive, instrumentation and control, project and spare parts. In OPEX, the following costs were considered: electricity, cooling water, maintenance and interests.
6.2.2 Cost Estimation of CAPEX and OPEX
The base Free On Board (FOB) purchase cost for the compressor section (CB) was estimated based on data from Garret (1989) and Walas (1988) from [55] which is a function of the total power requirement (PCin horsepower).
CB=exp(7.5800 + 0.80[ln(PC)]) (6.7) Since there is a considerable amount of water in the CO2stream, the compressors’ material has to be stainless steel, hence, a material factorFM (2.5 for stainless steel) is added to correct the cost (CP in US $ of 2006). This equation is valid for centrifugal compressors driven by an electric motor for a range ofPC from 200 to 30000 Hp.
CP =FMFCCB (6.8)
Considering the experience of compressor manufacturers that designs compressors specifically for CO2, another correction factorFCwas added to the cost function in order to obtain more accurate costs.
For the electric drive and the cooler cost, a similar procedure was performed where the cost functions came from [56]. For the electric drive, the cost function refers to the cost of a Variable Frequency Drive (VFD) motor and for the cooler refers to a shell and tube heat exchanger with the material of both sides being stainless steel and it’s valid for pressures up to 31.1 bar.
CDrive= 668.16PC+ 2049.3 (6.9)
CCooler = 330.4A+ 21025 (6.10)
WhereCDrive and CCooler (both in US$ of 2002) are the costs for the electric drive and cooler respectively,PCis the total duty of the electric drive in kW andAis the heat transfer area of the cooler in m2(valid for heat transfer areas between 100 and 1000 m2).
It was considered that only the compressors and the coolers are part of the base equipment, the instrumentation and control, project and spare parts costs were calculated as a percentage of base equipment cost. According to [56], the instrumentation and control can vary between 5 and 30%,
For the instrumentation and control, since the compressors have both surge and pressure controls and the coolers have level and temperature controls (at least), the compression train has a significantly high control density, so a 30% was adopted in this prediction.
For the project cost and spare parts, a percentage of 50% was used for their estimation because, on one hand, the compression of CO2 in Carbon Capture and Storage (CCS) is relatively a recent technology, increasing the project cost and, on the other hand, the compressors are maintenance intensive, requiring many spare parts.
The electricity price was obtained from [27] (0.05 US$/KWh of 2011) and the cooling water price was taken from [55] (0.02 US$/m3of 2007). Although this value seems relatively high when compared with the typical price in Portugal of 0.015 euros per kWh, the reason for this is that the capture cost is already taken in count in this price, making it higher.
Maintenance was estimated by assuming a 10% of CAPEX, which is the biggest value of the typical range suggested by [56], since compressors are know for being high maintenance intensive.
Finally, the interest cost was calculated by using a fixed tax of 7% of a 65% loan of CAPEX (typical value from [56]).
Note that, all the costs in the model were calculated to a reference year of 2011 using the Chemical Engineering Cost Factor Indexes (CEPI).
Table 6.1:Chemical Engineering Plant Cost Indexes.
year CEPI 2002 396 [57]
2006 500 [58]
2007 635 [59]
2011 699 [60]
6.3 Superstructure
Since the number of compressors is going to be determined by the optimization, the previous flow- sheet in figure 5.1 couldn’t be used as base structure to solve the optimization problem. gPROMS needs to have the mathematical structure of the problem fixed, in other words, the number of com- pressors would change through the optimization and therefore the number of equations and variables, provoking an error in gPROMS. So, the only way to prevent this is to fix the maximum number of com- pressors and use a by-pass strategy to determine the cost and the rest of the outputs.
To accomplish that, a superstructure that includes this feature was created:
Figure 6.1:Compression train superstructure topology in gPROMS environment.
Based on the usual trains designed for CO2 compression, a maximum number of 8 compressors was assumed for this superstructure, where the circle models between each compressor and cooler (calledBinary Splitter) are accessory models that decide if the stream flows to the next compressor or to the end of the train.
The Binary Splitter model only has one specification which is the by-pass fraction (yk). As the name implies, this variable represents the fraction of the inlet stream that is by-passed up to the end. So, the only thing the model does is determine how much of the stream is by-passed and how much flows to the next compressor. In a case where this variable is binary (0 or 1), it allows in the optimization to decide rather the CO2flows to the next compressor or flows directly to the end of the train, in other words, rather to use the next compressor or not.
However, this first mathematical approach proved to be inefficient due to the high number of unre- alistic possibilities during the relaxed NLP solver likeyk = [0,0,0.5,0,0.5,0,0,0]. Also, after the firstyk
becomes 1, the value of the following components of the array becomes irrelevant and, without any rule to set them, can lead to larger wasted simulation time (yk= [0,0,0,1,0,0,0,0] is analogous toyk = [0,0,0,1,1,0,0,0] oryk = [0,0,0,1,0,1,1,0]).
To avoid this situation, the final decision variable used is defined in gPROMS as a Special Ordered Set 1 (SOS-I). This variable is a set of binary variables where not more than one may have a non-zero value called ”flip” variable (zk). By definingzkas:
z1= 0 (6.11)
zk=yk−yk−1 ∀k= 2, . . . , Ns (6.12) Saying that a compression train has 5 compressor sections is the same as sayingy5= 1, translated to zk = [0,0,0,0,1,0,0,0] and leading toyk = [0,0,0,0,1,1,1,1] in the superstructure. This prevents the NLP solver having to optimize ”dry units” with zero flows which are temporarily turned off in the superstructure. Finally, the effect of non-convexities is reduced by special modeling techniques and the process units are modified to ensure that nonzero-flows are attained when the binary variables are set to zero.
Note that theCompressorSectionmodel is in design mode, where the user specifies the pressure ratio, and the CoolerKODrum model is also in design mode (water cooled option), where the user specifies the pressure drop and outlet temperature on both sides of the cooler as well as the heat
In table6.2 there’s a summary of all the inputs and key outputs of the system:
Table 6.2:Inputs and Key outputs of the compression train superstructure.
Inputs Key Outputs
Ns Cooler CO2pressure drop Total train duty Total cost
Ncoolers Cooler cw pressure drop Polytropic efficiency(k) CAPEX
pratio(k) Heat transfer coefficient Diameter(k) OPEX
CO2inlet conditions Cooling water temperature Cooling water flowrate Stream variables Cooler outletTCO2 Cooling water pressure Heat transfer area FinalPdisch
Cooler outletTcw Speed drive Tdischof compressor(k)
Further analysis of the number of Degrees of Freedom (DOF) will be done in the next sections.
6.4 Simplifications and Assumptions
In order to perform the first attempt to optimize the compression train, it was required to assume some simplifications and assumptions, in order to only have the number of compressors and pressure ratio as design variables.
The first assumption was that the CO2stream comes from a typical pre-combustion capture plant, so that the composition of the inlet CO2 stream used to solve to problem was the one from the IEA GHG report [27], presented in table 6.4.
Another assumption was that the pressure ratio is constant through the train, i.e., the pressure ratio of each compression section is the same. Thereasons for this assumption are that, on one hand, the systems becomes much simpler because it only has two degrees of freedom and, on another hand, it leads to the minimum ideal work (as described in section 2.3.2.
pratio(k) =pratio(k−1) ∀k= 2, . . . , Ns (6.13)
Since intercooling reduces the amount of compression work required, there was no decision if there is intercooling between sections or not. It was assumed that after each compressor sections there is always a cooler and knock-out drum to cool the CO2stream and remove water droplets. Ad- ditionally, this avoids having another by-pass strategy which would rise the complexity of the problem a lot. This implies that, in table 6.3, Ns= Ncoolers.
In the IEA GHG report from [27] there was no reference to what cooling utility was used in the intercoolers. So, since cooling water is the most common and economical one, it was assumed in this thesis a cooling water at 1 bar and 8oC, due to the low temperature specified for the outlet CO2
stream from the cooler (see table 6.3).
As for the other inputs described in tables 6.2, they were fixed and it was given the correspondent value from [27]:
Table 6.3:List of values of the input specification variables.
Input variable Value
Ns To be optimized
Ncoolers To be optimized
pratio(k) To be optimized
CO2inlet flowrate 58 kg/s CO2inlet temperature 268.15 K
CO2inlet pressure 1.2 bar Cooler outlet TCO2 292.15 K
Cooler outlet Tcw 290.15 K Cooler CO2pressure drop 0.2 bar
Cooler cw pressure drop 0.1 bar Heat transfer coefficient 400 W m−2K−1 Cooling water temperature 281.15 K
Cooling water pressure 1 bar
Speed drive 80 Hz
Where the CO2inlet streams’ composition has the typical composition of a pre-combustion capture plant:
Table 6.4:Inlet CO2stream composition of the superstructure.
Component Mass fraction % (kg/kg)
H2 0.04
CO2 99.77
H2O 0.17
CO 0.01
H2S 0.02
Finally, the compressors were designed on a section basis, i.e., there was no consideration about the number of stages per section. The design parameters, like efficiency and diameter, were the average value for the whole section. One of the main reasons to make this assumption was that the implementation of a stage design was still in development by the Energy Technologies Institute (ETI) team during the writing-up of this document. Another reason for this simplification is because optimiz- ing the number of stages per section would exponentially increase the complexity of the MINLP.
In this formulation, the unknown variables are the pressure ratio of each compressor (assuming they all have the same value) and the eight binary variables of zk (where their sum has to be 1), so the number of DOF of the system is eight.
DOF= 1 +Nsections−1 = 1 + 8−1 = 8 (6.14) If the number of stages was to be optimized, assuming the limit of nine stages per section for centrifugal compressors and every stage in each section would have the same pressure ratio; the pressure ratio of the stages and the number of stages would have to be optimized for each section.
So, the number of DOF would be:
DOF= 1 + (Nsections−1)×(1 + 1 + (Nstages−1)) = 1 + 7×(2 + 8) = 71 (6.15) A problem like this is far to complicated to solve at this stage and therefore is not the goal of this